The graph of a sinusoidal function intersects its midline at $(0,1)$ and then has a maximum point at $\left(\dfrac{7\pi}{4},5\right)$. Write the formula of the function, where $x$ is entered in radians. $f(x)=$
Explanation: The strategy First, let's use the given information to determine the function's amplitude, midline, and period. Then, we should determine whether to use a sine or a cosine function, based on the point where $x=0$. Finally, we should determine the parameters of the function's formula by considering all the above. Determining the amplitude, midline, and period The midline intersection is at $y={1}$, so this is the midline. The maximum point is $4$ units above the midline, so the amplitude is ${4}$. The maximum point is $\dfrac{7\pi}{4}$ units to the right of the midline intersection, so the period is $4\cdot \dfrac{7\pi}{4}={7\pi}$. [Why did we multiply by 4?] Determining the type of function to use Since the graph intersects its midline at $x=0$, we should use the sine function and not the cosine function. This means there's no horizontal shift, so the function is of the form $a\sin(bx)+d$. [How do we know that?] Determining the parameters in $a\sin(bx)+d$ Since the midline intersection at $x=0$ is followed by a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${4}$, so $|a|={4}$. Since $a>0$, we can conclude that $a=4$. The midline is $y={1}$, so $d=1$. The period is ${7\pi}$, so $b=\dfrac{2\pi}{{7\pi}}=\dfrac{2}{7}$. The answer $f(x)=4\sin\left(\dfrac{2}{7}x\right)+1$